The proposed work continues in line with our efforts to describe the cognitive abilities of preschoolers. The focus is on the nature of concepts, number knowledge, and communication skills in the preschool-aged child. Research on the nature of concepts is designed to determine whether the nature of concepts is different for different classes of properties or objects, e.g., color versus animals. The number work focuses on our theory of the nature and acquisition of number concepts and arithmetical reasoning abilities. The communication work focuses on a description of how beginning language learners acquire rules of conversation, come to alter their language when talking to different kinds of listeners (e.g., very young children versus adults), and knowledge about their various listeners' needs, capacities, and perspectives. Our combined effort to determine what cognitive abilities preschoolers have and how these serve as building blocks for the further acquisition of knowledge can provide a body of information for educators who seek to evolve educational programs for preschool-aged children.